Ruin probability in the presence of risky investments

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Ruin probability in the presence of risky investments

Serguei Pergamenchtchikov, Zeitouny Omar

To cite this version:

Serguei Pergamenchtchikov, Zeitouny Omar. Ruin probability in the presence of risky investments.

2008. �hal-00532597�

Ruin probability in the presence of risky investments. ^∗

Serguei Pergamenshchikov

Omar Zeitouny

Abstract

We consider an insurance company in the case when the premium rate is a bounded non-negative random function c

and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brow- nian motion with mean return a and volatility σ > 0. If β := 2a/σ

− 1 > 0 we find exact the asymptotic upper and lower bounds for the ruin proba- bility Ψ(u) as the initial endowment u tends to infinity, i.e. we show that C

u

≤ Ψ(u) ≤ C

u

for sufficiently large u. Moreover if c

= c

e

with γ ≤ 0 we find the exact asymptotics of the ruin probability, namely Ψ(u) ∼ u

. If β ≤ 0, we show that Ψ(u) = 1 for any u ≥ 0.

MJS: primary 62P05; 60J25; G22; G23

Keywords: Risk process; Geometric Brownian motion; Ruin probability

1 Introduction

It is well known that the analysis of activity of an insurance company in conditions of uncertainty is of great importance. Starting from the classical papers of Cram´er and Lundberg which first considered the ruin problem in stochastic environment, this subject has attracted much attention. Recall that, in the classical Cram´er–

Lundberg model satisfying the Cram´er condition and, the positive safety loading

∗

This work is partly supported by the German Science Foundation (Deutsche Forschungsgemein- schaft) through the Graduiertenkolleg ”Angewandte Algorithmische Mathematik” at Munich Univer- sity of Technology and by RFFI - Grant 04-01-00855.

†

Laboratoire de Math´ematiques Rapha¨el Salem, UMR 6085 CNRS-Univ. de Rouen Avenue de l’Universit´e, BP.12, 76801 Saint Etienne du Rouvray, France, e-mail:

Serge.Pergamenchtchikov@univ-rouen.fr

‡

Laboratoire de Math´ematiques Rapha¨el Salem, UMR 6085 CNRS-Univ. de Rouen Avenue de

l’Universit´e, BP.12, 76801 Saint Etienne du Rouvray, France, e-mail: Omar.Zeitouny@univ-rouen.fr

assumption, the ruin probability as a function of the initial endowment decreases exponentially (see, for example, Mikosch [12]). The problem was subsequently extended to the case when the insurance risk process is a general L´evy process (see, for example, Kl¨uppelberg et al. [10] for details).

More recently ruin problems have been studied in application to an insurance company which invests its capital in a risky asset see, e.g., Paulsen [14], Kalsh- nikov and Norberg [9], Frolova et al. [5] and many others.

It is clear that, risky investment can be dangerous: disasters may arrive in the period when the market value of assets is low and the company will not be able to cover losses by selling these assets because of price fluctuations. Regulators are rather attentive to this issue and impose stringent constraints on company port- folios. Typically, junk bonds are prohibited and a prescribed (large) part of the portfolio should contain non-risky assets (e.g., Treasury bonds) while in the re- maining part only risky assets with good ratings are allowed. The common notion that investments in an asset with stochastic interest rate may be too risky for an insurance company can be justified mathematically.

We deal with the ruin problem for an insurance company investing its capital in a risky asset specified by a geometric Brownian motion

dV

= V

(adt + σdw

) , (1.1)

where (w

, t ≥ 0) is a standard Brownion motion and a > 0, σ > 0.

It turns out that in this case of small volatility, i.e. 0 < σ

< 2a, the ruin probability is not exponential but a power function of the initial capital with the exponent β := 2a/σ

− 1. It will be noted that this result holds without the requirement of positive safety loading. Also, for large volatility, i.e. σ

> 2a, the ruin probability equals 1 for any initial endowment. These results have been obtained under various conditions in [14, 9, 5].

Additionally, a large deviations limiting theorems for describing the ruin prob- ability was obtained by Djehiche [3] and Nyrhinen [13]. Gaier et al. [6] studied the optimal investment problem for an insurance company.

In all these papers the premium rate was assumed to be constant. In practice this means that the company should obtain a premium with the same rate continu- ously. We think that this condition is too restrictive and it significantly bounds the applicability of the above mentioned results in practical insurance settings.

The goal of this paper is to consider the ruin problem for an insurance company

for which the premium rate is specified by a bounded non-negative random func-

tion c

. For the given problem, under the condition of small volatility, we derive

exact upper and lower bounds for the ruin probability and in the case of exponential

premium rate, i.e. c

= e

with γ ≤ 0, we find the exact asymptotics for the ruin

probability. Particularly, we show that for the zero premium rate, i.e. γ = −∞ , the asymptotic result is the same as in the case −∞ < γ < 0.

Moreover, in this paper we show that in the boundary case, i.e. σ

= 2a, the company goes bankrupt with probability 1 for any bounded function c

.

Indeed, an upper bound for the ruin probability for the random function c

in the small volatility case is obtained also by Ma and Sun [11].

The paper is organised as follows. In the next section we give the main results.

In Section 3 we give the necessary results about the tails of solutions of some linear random equation which we apply to study the ruin problem. In Section 4 we obtain the upper bound for the ruin probability and in Section 5 we find the corresponding lower bound. In Section 6 we consider the exponential premium income rate case.

In Section 7 we study some ergodic properties for an autoregressive process with random coefficient. in Section 8 we consider the large volatility case.

2 Basic results

Let us consider a process X = X

of the form X

= u + a

Z

X

ds + σ Z

0

X

dw

+ Z

0

c

ds −

Nt

X

i=1

ξ

, (2.1) where a ≥ 0 and σ ≥ 0 are arbitrary constants, w is a Brownion motion, N is a Poisson process with intensity α > 0 and (ξ

, i ∈ N ) are i.i.d. positive random variables with common a distribution F. Moreover, we assume that w, N , (ξ

) are independent and the filtration is defined as F

= σ n

w

, N

, P

i=1

ξ

, 0 ≤ s ≤ t o . Furthermore, c

= c(t, X) is a bounded non-negative ( F

) - adapted function (i.e., 0 ≤ c

≤ c

) such that Eq. (2.1) has an unique strong solution (see chapter 14 in [8]).

Let ς

:= inf { t : X

< 0 } (the time of ruin), Ψ(u) := P (ς

< ∞ ) (the ruin probability). The parameter values a = 0, σ = 0, c

= c, correspond to the Cram´er–Lundberg model for which the risk process is usually written as X

= u + c t − P

i=1

ξ

. In the considered version (of non-life insurance) the capi-

tal evolves due to a continuously incoming cash flow with rate c > 0 and outgoing

random payoffs ξ

at times forming an independent Poisson process N with inten-

sity α. For the model with positive safety loading and F having a ”non-heavy” tail,

the Lundberg inequality provides encouraging information: the ruin probability

decreases exponentially as the initial endowment u tends to infinity. Moreover, for

exponentially distributed claims the ruin probability admits an explicit expression,

see [1] or [12].

We study here the case σ > 0 with a general random adapted bounded function c

. In this case Eq. (2.1) describes the evolution of the capital of an insurance company, which is continuously reinvested into an asset with the price following a geometric Brownian motion (1.1).

Let β := 2a/σ

− 1. To write the upper bound for the ruin probability we define the function :

J (β) = 2α σ

β

1

+ j

(β) 1

+ j

(β) 1

, (2.2)

where j

(β) = β (1 + ̺

), j

(β) = β 2

(1 + ((1 + ̺)

− 1)

) and

̺ = ̺(β) = (β − 1)σ

/2α.

Theorem 2.1. If β > 0 and E ξ

< ∞ , then lim sup

u

Ψ(u) ≤ C

(β), where C

(β ) = J(β) E ξ

.

The proof of this theorem is given in Section 4.

Theorem 2.2. If β > 0 and E ξ

< ∞ for some δ > 0, then there exists a constant 0 < C

< ∞ such that lim inf

u

Ψ(u) ≥ C

.

This result is proved in Section 5. The following theorem gives the exact asymp- totics for the exponential function c

.

Theorem 2.3. Assume that c

= c

exp { γt } with −∞ ≤ γ ≤ 0. If β > 0 and E ξ

< ∞ for some δ > 0, then there exists a constant 0 < C

< ∞ such that lim

u

Ψ(u) = C

. Moreover, the constant C

is the same for any

−∞ ≤ γ < 0.

This result is proved in Section 6. Now we consider the large volatility case, i.e.

β ≤ 0.

Theorem 2.4. Assume that the distribution of ξ

has not a finite support, i.e.

P (ξ

> z) > 0 for any z ∈ R . If β ≤ 0 and E ξ

< ∞ for some δ > 0, then Ψ(u) = 1 for any u ≥ 0.

Remark 2.5. This theorem has been proved by Paulsen in [14] for a constant premium rate, i.e. for c

= c

= const.

The key idea in the proofs of Theorem 2.1 and Theorem 2.2 is based on the fact

that the function Ψ(u) may be estimated by the tails of solutions of some linear

random equations. In the next section we study the asymptotic behaviour of those

tails.

3 Tails of solutions of random equations

This Section contains some results from the general renewal theory developed by Goldie [7] for some random equations. We consider the following two random equations

R

= Q + M R , R is independent of (M, Q) (3.1) (

= denoting equality of probability laws) and

R

= Q + M (R

)

, R

independent of (M, Q) , (3.2) where (a)

= max(a, 0).

We start with some preliminary conditions for the random variable M which are studied by Goldie (see Lemma 2.2 in [7]).

Lemma 3.1. Let M ≥ 0 be a random variable such that, for some β > 0

E M

= 1 , E M

(log M)

< ∞ (3.3) and the conditional law of log M, given M 6 = 0, be non-arithmetic. Then −∞ ≤ log E M < 0 and 0 < µ := E M

log M < ∞ .

The following result from [7] specifies the tail behaviour of R.

Lemma 3.2. (Theorem 4.1 in [7]) Let M be a random variable satisfying the conditions of Lemma 3.1 for some β > 0 and Q be a positive random variable for which E Q

< ∞ . Then there is a unique law for R satisfying (3.1) such that

u→

lim

u

P (R > u) = c

, (3.4) where c

= E

(Q + M R)

− (M R)

/β µ and µ = E M

log M . Now we study the tail of R

.

Lemma 3.3. Let M ≥ 0 be a random variable satisfying the conditions of Lemma 3.1 for some β > 0. Assume also that the distribution of M is absolutely continuous with respect to Lebesgue measure and there exists δ > 0 such that

E M

< ∞ (3.5)

and for any x ∈ R

E M

6 = 1 , (3.6)

where i = √

− 1. Then under the condition

E | Q |

< ∞ (3.7)

for some δ > 0 there is a unique law for R

satisfying (3.2) such that there exists lim

u

P (R

> u) = c

and 0 < c

< ∞ .

This lemma follows directly from Theorem 6.3 in [7] and Theorem 2 in [13].

4 Upper bound for the ruin probability

Let τ

be the instant of n-th jump of N and let θ

:= τ

− τ

with τ

:= 0. We define the discrete-time process S = S

with S

:= X

. Since ruin may occur only when X jumps downwards, Ψ(u) = P (T

< ∞ ), where

T

:= inf { n ≥ 1 : S

< 0 } . (4.1) Therefore to obtain asymptotic properties of T

as u → ∞ we need to study the process (S

). First of all, we need to find a recurrence equation for this sequence.

We start with resolving of Eq. (2.1). For this we introduce the process (φ

)

which satisfies the following stochastic differential equation

dφ

= a φ

dt + σ φ

dw

+ c

dt , φ

= x . The Ito formula implies that φ

= e

x + R

s

e

c

du, where

h

= κ t + σw

, κ = a − σ

/2 and t ≥ s. Moreover we can represent Eq. (2.1) for τ

≤ t < τ

in the following way

X

= S

+ a Z

τn−1

X

ds + σ Z

τn−1

X

dw

+ Z

τn−1

c

ds = φ

= e

S

+ Z

τn−1

e

c

du .

Therefore S

= X

= φ

n

− ξ

. From this we obtain the following random recurrence equation for (S

)

S

= λ

S

+ ζ

, S

= u (4.2) with λ

= exp { σ w

n

+ κθ

} and ζ

= η

− ξ

. Here w

= w

n−1

− w

n−1

and η

= R

0

c

e

du with c

:= c

. By resolving (4.2) we find the following representation for (S

)

S

= E

u + E

X

k=1

E

ζ

, E

=

n

Y

k=1

λ

. (4.3)

Moreover, taking into account here that ζ

≥ − ξ

we obtain that S

≥ E

(u − Y

), where

Y

= Q

+

n

X

k=2

Q

k−1

Y

j=1

M

, M

= λ

, Q

= ξ

/λ

. (4.4) Notice that (M

) are i.i.d. random variables such that for q ∈ ]0, β]

E M

= E λ

= 2α

2α + (β − q) q σ

≤ 1 . (4.5) Therefore, there exists 0 < δ < min(1, β) for which ρ = E M

< 1 and

E



 X

k≥2

Q

k−1

Y

j=1

M





δ

≤ X

k≥2

E



Q

k−1

Y

j=1

M





δ

= E Q

X

k≥2

ρ

< ∞ ,

i.e. the series P

k≥2

Q

Q

j=1

M

is finite a.s. It means that the sequence (Y

) have a finite limit

n

lim

Y

= Q

+

+∞

X

k=2

Q

k−1

Y

j=1

M

= Y

= R < ∞ a.s. (4.6) Taking into account that the sequence (Y

) in (4.4) is increasing we can estimate S

as

S

≥ E

(u − R) (4.7)

and by (4.1) we get that P (T

< ∞ ) ≤ P (R > u). Therefore, to obtain the upper bound for the ruin probability we investigate the tail behaviour of R as u → ∞ . To this end, first notice that we may represent R in the following form

R = Q

+ M

R

, (4.8)

where the random variable R

= Q

+ P

Q

j=2

M

Q

has the same distri- bution as R and is independent of (Q

, M

). Thus the random variable R satisfies Eq. (3.1). We show that

u

lim

u

P (R > u) = C

, (4.9) where C

= 2α E ((ξ

+ R)

− R

)/β

σ

.

To show (4.9) we need to check the conditions of Lemma 3.2 for the random variables (M

) and (Q

) defined in (4.4). The first property in (3.3) follows directly from (4.5) for q = β. Now we show the second. By definition of M

we have

E M

(log M

)

= E e

( − σ w

− κ θ

) 1

1−κ θ1≥0}

≤ σ E | w

| e

+ κ E θ

e

.

Taking into account that (w

) is independent of (θ

), the last term in this inequality equals

σ 1

√ 2π E p θ

Z

−∞

| z | e

dz + κ E θ

, i.e. E M

(log M

)

≤ (βσ

+ κ) E θ

+ σ p

2/π E √

θ

< ∞ . In similar way we calculate µ = E M

log M

= βσ

/2α. Moreover, E Q

= E ξ

< ∞ . Therefore, by making use of Lemma 3.2 we get the limiting relationship (4.9) which implies that lim sup

u

Ψ(u) ≤ C

. Thus, to finish the proof we need to show the inequality C

≤ C

(β). Indeed, if 0 < β ≤ 1, then

E ((ξ

+ R)

− R

) ≤ E ξ

and, therefore, in this case C

≤ C

(β). If β > 1, then, taking into account the inequality a

− b

≤ β (a − b) a

(0 < b < a), we obtain that C

≤ 2α E ξ

(ξ

+ R)

/βσ

. This implies that for 1 < β ≤ 2,

C

≤ 2α

β σ

( E ξ

+ E ξ

E R

) ≤ 2α

β σ

( E ξ

+ ( E ξ

)

E R

) . (4.10) Since by (4.5) we have E M

< 1, therefore by making use of (4.8) and taking into account that ( E M

)

− 1 = ̺ (̺ is defined in (2.2)) we can estimate E R

as

E R

≤ E Q

1 − E M

= E ξ

E M

1 − E M

≤ 1

̺ ( E ξ

)

.

Thus, from this and (4.10), we obtain that C

≤ C

(β) for 1 < β ≤ 2. Let us consider now the case β > 2. In this case we estimate C

as

C

≤ 2

α

βσ

( E ξ

+ E ξ

R

) ≤ 2

α

β σ

( E ξ

+ ( E ξ

)

E R

) . (4.11) We set k R k

= ( E R

)

with q = β − 1. Taking into account that the random variables R

and M

are independent in (4.8), we obtain that

k R k

= k M

R

+ Q

k

≤ k M

k

k R

k

+ k Q

k

,

i.e. k R k

≤ k Q

k

(1 − k M

k

)

= k ξ

k

(( k M

k

)

− 1)

. From this, we find

E R

≤

(1 + ̺)

− 1

( E ξ

)

.

Applying this inequality to (4.11), one obtains C

≤ C

(β) for β > 2. This

implies Theorem 2.1.

5 Lower bound for the ruin probability

In this section we prove Theorem 2.2. First, notice that the identity (4.3) implies S

≤ S

:= E

u + E

n

X

k=1

E

ζ

, (5.1)

where ζ

= η

− ξ

with η

:= c

R

0

e

du. Therefore, denoting T

= inf { n ≥ 1 : S

< 0 } we obtain

Ψ(u) = P (T

< ∞ ) ≥ P (T

< ∞ ) , (5.2) for any u > 0. Setting Q

= (ξ

− η

)/λ

in (5.1), we represent S

in the following form S

= E

(u − Y

), where Y

= Q

and for n ≥ 2,

Y

= Q

+ M

Q

+ · · · +

n−1

Y

j=1

M

Q

. (5.3)

Therefore, for any u > 0,

P (T

< ∞ ) = P (R

> u) , (5.4) where R

= sup

Y

. To study the tail behaviour of R

we need to obtain the renewal equation for R

. To this end we rewrite Y

as Y

= Q

+ M

Z

with Z

= Q

and Z

= Q

+ M

Q

+ · · · + Q

j=2

M

Q

for n ≥ 2. By denoting R

:= sup

Z

we get that R

= Q

+ M

(R

)

. Note that the random vector (Z

, . . . , Z

) has the same distribution as (Y

, . . . , Y

) for any n ≥ 2, i.e. R

has the same distribution as R

also. Moreover, taking into account that R

is independent of (Q

, M

), we deduce that R

satisfies the random Eq. (3.2). We show now that

u

lim

u

P (R

> u) = C

> 0 . (5.5) To prove this we check the conditions of Lemma 3.3. First, notice that (4.5) implies (3.5) for any 0 < δ < p

α

+ β

/4 − β/2 with α

= 2α/σ

. It easy to see that for such δ and any x ∈ R in this case E M

6 = 1. Now we verify (3.7). Writing q = β + ǫ with ǫ > 0, we obtain

E | Q

|

≤ const ( E M

E ξ

+ E (η

M

)

) .

By the conditions of Theorem 2.1 and (4.5) the first term in this inequality is finite

for sufficiently small ǫ. Moreover, we prove that there exists ε > 0 such that

E (η

M

)

< ∞ . Indeed, setting w

= sup

( − w

−

s) we get E (η

M

)

= (c

)

E

Z

0

e

du

≤ (c

)

E θ

e

= (c

)

α Z

0

t

E e

e

dt .

The last intergal we estimate as Z

0

t

E e

e

dt ≤ 2

α K

E e

,

where K

= sup

(t

e

) and τ is an exponential random variable with the parameter α/2 independing on (w

)

. Moreover, taking into account that the random variable w

is exponential (see, for example, [2] p. 197) we find that

K

= E e

=

√ ασ

+ κ

+ κ

√ ασ

+ κ

− κ − εσ

< ∞

for 0 < εσ

< √

ασ

+ κ

− κ. Therefore we get

E (η

M

)

≤ 2(c

)

K

K

. (5.6) Now (5.5) follows from Lemma 3.3. Hence Theorem 2.2.

6 Exact asymptotics for the ruin probability

In this subsection we prove Theorem 2.3. For γ = 0, the theorem follows from (5.5). Therefore we assume −∞ < γ < 0. In this case Eq. (4.2) has the following form

S

= E

u + E

X

k=1

E

(c

η ˜

− ξ

) , (6.1) where c

= c

= c

exp { γτ

} and η ˜

= R

0

e

du. We set Y ˜

:=

P

k=1

E

c

η ˜

= P

k=1

Q

j=1

M ˜

Q ˜

with Q ˜

= c

M

η ˜

and

M ˜

= e

M

. Taking into account that ( ˜ Y

) is an increasing sequence, we put R ˜ = ˜ Y

= lim

n→∞

Y ˜

= X

k=1 k−1

Y

j=1

M ˜

Q ˜

a.s.

Notice now that this random variable satisfies the following identity in law R ˜

= ˜ Q + ˜ M R , ˜

where Q ˜

= ˜ Q

, M ˜

= M ˜

and R ˜ is independent of ( ˜ Q, M ˜ ). Moreover, for q = ˜ β = β − 2γ/σ

we get E M ˜

= α(α + ( ˜ β − q) q σ

/2)

= 1 and simi- larly to (5.6) we can show that E Q ˜

< ∞ . Therefore, Lemma 3.2 implies that lim

u

P ( ˜ R > u) < ∞ . Thus, by (4.9)

u

lim

P ( ˜ R > u)

P (R > u) = 0 . (6.2) Now we study the stopping time (4.1) in our case. First, by (6.1) we may write T

as

T

:= inf { n ≥ 1 : S

< 0 } = inf { n ≥ 1 : Y

> u + ˜ Y

} , (6.3) where Y

is defined in (4.4).

Recall that, R = Y

= lim

Y

a.s. and R ˜ = ˜ Y

= lim

Y ˜

a.s..

Therefore from (6.3) it follows that P (R > u + ˜ R , T

= ∞ ) = 0. Taking this into account, it easy to deduce the following equality

P (T

< ∞ ) = P (Y

> u + ˜ Y

) = P (R > u + ˜ Y

) . (6.4) From here we obtain for any δ > 0,

P (T

< ∞ ) ≥ P (R > u + ˜ Y

, Y ˜

≤ δu) ≥ P (R > (1 + δ)u, Y ˜

≤ δu)

= P (R > (1 + δ)u) − P (R > (1 + δ)u , Y ˜

> δ u)

≥ P (R > (1 + δ)u) − P ( ˜ R > δ u) . The limiting relationships (4.9) and (6.2) imply that

lim inf

u→+∞

P (T

< ∞ )/ P (R > u) ≥ 1 .

Moreover, by (6.4) we obtain P (T

< ∞ ) ≤ P (R > u) for any u > 0. Thus

u

lim

P (T

< ∞ )/ P (R > u) = 1 .

If γ = −∞ , i.e. c

= 0, then Y ˜

= 0 for all n ∈ N and , hence, P (T

< ∞ ) =

P (R > u). Therefore (4.9) implies this theorem in this case.

7 Erdodic properties for the random coefficient autore- gressive process

To show Theorem 2.4 we need to use some ergodic properties of the special au- toregressive process with random coefficients (5.1). In this section we study the ergodic properties for a general scalar autoregressive process with random coeffi- cient

x

= a

x

+ b

, n ≥ 1 , (7.1) where x

is some fixed constant and (a

, b

) is i.i.d. sequence of random variables in R

.

Proposition 7.1. Assume that there exists 0 < δ ≤ 1 such that ρ = E | a

|

< 1 and E | b

|

< ∞ . Then for any bounded uniformly continuous function f

P − lim

N→∞

N

N

X

n=1

f (x

) = E f (x

) , (7.2) where x

= P

k=1

π

b

with π

= 1 and π

= Q

j=1

a

for k ≥ 1.

Proof. First we show that the series in the definition of x

converges in probabil- ity. Indeed, E | P

k=n

π

b

|

≤ E | b

|

P

k=n

ρ

. It means that the series P

k≥1

π

b

converges in L

and hence in probability. Now we fixe some m ≥ 1 and, for n ≥ m, we set x

(m) = P

k=n−m+1

b

Q

j=k+1

a

. Notice that x

(m) is mesurable with respect to σ { a

, . . . , a

, b

, . . . , b

} . Therefore for any 0 ≤ d < m the the sequence (x

)

is i.i.d. and by the law of large numbers for any fixed m ≥ 1 and 0 ≤ d < m

p

lim

p

p

X

k=1

f (x

(m)) = E f (x

(m)) a.s. , (7.3)

where x

(m) = P

k=1

b

Q

j=k+1

a

= P

k=1

b

π

. Therefore

m

lim

E f (x

(m)) = E f (x

) . (7.4) We show now that for any ǫ > 0

m

lim

sup

N≥m

P ( ∆(N, m) > ǫ ) = 0 , (7.5) where ∆(N, m) = N

P

n=m

| f (x

) − f (x

(m)) | .

We put x

(m) = x

− x

(m) = x

Q

k=n−m+1

a

. Taking into account that there exists some L

< ∞ such that for any n ≥ 1

E | x

|

= E | x

n

Y

k=1

a

+

n

X

k=2

b

n

Y

j=k+1

a

|

≤ | x

|

ρ

+ E | b

|

n

X

k=2

ρ

≤ L

, we get sup

E | x

(m) |

≤ L

ρ

.

Let us choose ǫ

> 0 for which sup

| f (x) − f (y) | ≤ ǫ/2. For such ǫ

we obtain that ∆(N, m) ≤ ǫ/2 + 2f

N

P

n=m

1

, where f

= sup

| f (x) | . Therefore by denoting ǫ

= ǫ/4f

we get that

P ( ∆(N, m) > ǫ ) ≤ P

N

X

n=m

1

> ǫ

N

! .

Applying here the Chebyshev inequality we find that P ( ∆(N, m) > ǫ ) ≤ 1

ǫ

N

N

X

n=m

P ( | x

(m) | ≥ ǫ

) ≤ L

1 ǫ

ǫ

ρ

. This implies (7.5). We put p = [N/m] ([a] is the whole part of a), i.e. N = pm +r with 0 ≤ r < m). For such p and r, we can write that

Ω

:=

1 N

N

X

n=1

f (x

) − E f (x

)

≤

1 N

pm−1

X

n=m

f (x

(m)) − E f (x

) + f

m + r

N + ∆(N, m) . Moreover, we can represent the last sum in this inequality as

pm−1

X

n=m

f (x

(m)) =

m−1

X

d=0 p−1

X

k=1

f (x

(m)) .

Therefore, from (7.3), we get that

N

lim

1 N

pm−1

X

n=m

f (x

(m)) = lim

p→∞

1 m

m−1

X

d=0

1 p

p−1

X

k=1

f (x

(m)) = E f (x

(m)) .

Finally, for ǫ > 0, we obtain that for any m ≥ 1 lim sup

N→∞

P ( Ω

> ǫ ) ≤ sup

N≥1

P ( |E f (x

) − E f (x

(m)) | + ∆(N, m) > ǫ ) .

The limiting relationships (7.4)–(7.5) imply (7.2).

8 Large volatility

In this section we prove Theorem 2.4. First, notice that if β < 0 then Proposition 4 in [5] implies that P (T

< ∞ ) = 1 for any u ≥ 0. Thus Theorem 2.4 for β < 0 directly follows from Inequality (5.2). We consider the critical case β = 0, i.e.

κ = 0 and λ

= e

with ν

= w

k

= w

k

− w

k−1

.

For this, we study the ergodic properties of the process (S

) defined in (5.1).

Notice that (5.1) implies that this process satifies the following random reccurence equation

S

= λ

S

+ ζ

, (8.1) where S

= u and ζ

is defined in (5.1).

Set t

= 0 and t

= inf { k > t

: P

j=tn−1+1

ν

< 0 } for n ≥ 1. It is easy to see that t

= P

j=1

ρ

, where (ρ

) is an i.i.d. sequence which has the same distribution as t

whose properties are well known, see XII. 7 theorem 1a in [4].

One can show, that for some constant 0 < c < ∞ , sup

n≥1

n

P (t

> n) ≤ c . (8.2) Set x

= S

n

. By (8.1) we obtain that for any n ≥ 1,

x

= a

x

+ b

, x

= u , (8.3) where a

= Q

j=1

λ

= exp { σ P

j=1

ν

} and b

=

ρn

X

k=1

(

ρn

Y

j=k+1

λ

) ζ

n−1+k

.

The sequence (a

, b

) is an i.i.d. sequence of random variables in R

. Moreover, E a

= E a

< 1. We will show that there exists r > 0 such that

E | b

|

< ∞ . (8.4)

First, notice that the definition of b

implies that | b

| ≤ P

k=1

| ζ

| . Moreover,

similarly to (5.6) we can show that there exists 0 < ǫ < 1 for which E | η

|

< ∞ .

Therefore, taking into account the condition of Theorem 2.4 ( E ξ

< ∞ for some

δ > 0) we get that there exists 0 < ǫ < 1 such that m

= E | ζ

|

< ∞ . To finish

the proof of inequality (8.4), note that, for such ǫ and for some fixed 0 < r < 1, by

making use of inequality (8.2) we obtain that E | b

|

≤ 1 + r

X

n=1

1 n

P (

t1

X

k=1

| ζ

| > n)

≤ 1 + r

∞

X

n=1

1 n

P (

ln

X

k=1

| ζ

| > n) + r

∞

X

n=1

1

n

P (t

> l

)

≤ 1 + r m

∞

X

n=1

l

n

+ r c

∞

X

n=1

1 n

l

.

Therefore, by putting l

= [n

], we obtain (8.4) for 0 < r < ǫ/5. Hence, by Proposition 7.1, the process (8.3) has the property (7.2) for some bounded uniform continuous function f .

For Eq. (8.3) we reprsent the random variable x

= P

k ≥ 1π

b

as x

:= Q

j=2

λ

(ς − ξ

), where ς is independent of ξ

. This implies that P (x

<

0) = P (ξ

> ς). Thus, by the condition on the distribution of ξ

we obtain that P (x

< 0) > 0. It means that for the function f

(x) = min(x

, 1) 1

we have E f

(x

) > 0 and by (7.2) there exists a sequence (n

) such that

lim

n

P

j=1

f

(x

) = E f

(x

) > 0 a.s. Therefore P (T

< ∞ ) = 1 and Theorem 2.3 follows directly from (5.2).

Acknowledgments

The authors thank Thomas Mikosch and Claudia Kl¨uppelberg for helpful re- marks and comments.

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